I'm fairly confused about the above, and haven't succeeded in rigorously defining a satisfying model from which I can compute the expected utility of a donation. My intuition is that the expected value of a donation should work out to the good done by the number of nets that can be purchased and distributed with that amount of money, but I can't justify it.

Suppose they distribute bed nets in batches of D, and your donation can buy K < D nets. There is a probability of K/D that your donation triggers another batch of D nets, hence expected number of nets = (K/D)*D = K. (By a slightly longer argument, this is still true even when K >= D.)

Note that by the same reasoning, cutting 1 minute off your journey to catch a train, when you have no knowledge of the train timetable, has an expected saving of 1 minute on your whole journey, regardless of how frequently the trains run.